Subgraph Enumeration in Massive Graphs

Abstract: We consider the problem of enumerating all instances of a given pattern graph in a large data graph. Our focus is on determining the input/output (I/O) complexity of this problem. Let $E$ be the number of edges in the data graph, $k=O(1)$ be the number of vertices in the pattern graph, $B$ be the block length, and $M$ be the main memory size. The main results of the paper are two algorithms that enumerate all instances of the pattern graph. The first one is a deterministic algorithm that exploits a suitable independent set of the pattern graph of size $1\leq s \leq k/2$ and requires $O\left(E^{k-s}/\left(BM^{k-s-1}\right)\right)$ I/Os. The second algorithm is a randomized algorithm that enumerates all instances in $O\left(E^{k/2}/\left(BM^{k/2-1}\right)\right)$ expected I/Os; the same bound also applies with high probability under some assumptions. A lower bound shows that the deterministic algorithm is optimal for some pattern graphs with $s=k/2$ (e.g., paths and cycles of even length, meshes of even side), while the randomized algorithm is optimal for a wide class of pattern graphs, called Alon class (e.g., cliques, cycles and every graph with a perfect matching).